A256554 -id:A256554 - cp's OEIS frontend

A074761 Number of partitions of n of order n.

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 9, 1, 4, 5, 1, 1, 12, 1, 27, 7, 6, 1, 81, 1, 7, 1, 54, 1, 407, 1, 1, 11, 9, 13, 494, 1, 10, 13, 423, 1, 981, 1, 137, 115, 12, 1, 1309, 1, 59, 17, 193, 1, 240, 21, 1207, 19, 15, 1, 47274, 1, 16, 239, 1, 25, 3284, 1, 333, 23, 3731, 1, 42109, 1, 19

Offset: 1

Vladeta Jovovic, Sep 28 2002

Order of partition is lcm of its parts.

a(n) is the number of conjugacy classes of the symmetric group S_n such that a representative of the class has order n. Here order means the order of an element of a group. Note that a(n) = 1 if and only if n is a prime power. - W. Edwin Clark, Aug 05 2014

			The a(15) = 5 partitions are (15), (5,3,3,3,1), (5,5,3,1,1), (5,3,3,1,1,1,1), (5,3,1,1,1,1,1,1,1). - _Gus Wiseman_, Aug 01 2018

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 1..4000 (first 1025 terms from Joerg Arndt)

Cf. A018818, A074351, A074752.
Main diagonal of A256067, A256554.
Cf. A000837, A074761, A285572, A290103, A305566, A316429, A316431, A316432, A316433, A317624.

A:= proc(n)
      uses numtheory;
      local S;
    S:= add(mobius(n/i)*1/mul(1-x^j,j=divisors(i)),i=divisors(n));
    coeff(series(S,x,n+1),x,n);
end proc:
seq(A(n),n=1..100); # Robert Israel, Aug 06 2014

a[n_] := With[{s = Sum[MoebiusMu[n/i]*1/Product[1-x^j, {j, Divisors[i]}], {i, Divisors[n]}]}, SeriesCoefficient[s, {x, 0, n}]]; Array[a, 80] (* Jean-François Alcover, Feb 29 2016 *)
Table[Length[Select[IntegerPartitions[n],LCM@@#==n&]],{n,50}] (* Gus Wiseman, Aug 01 2018 *)

pr(k, x)={my(t=1); fordiv(k, d, t *= (1-x^d) ); return(t); }
a(n) =
{
    my( x = 'x+O('x^(n+1)) );
    polcoeff( Pol( sumdiv(n, i, moebius(n/i) / pr(i, x) ) ), n );
}
vector(66, n, a(n) )
\\ Joerg Arndt, Aug 06 2014

Coefficient of x^n in expansion of Sum_{i divides n} A008683(n/i)*1/Product_{j divides i} (1-x^j).

A074064 Number of cycle types of degree-n permutations having the maximum possible order.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 2, 4

Offset: 0

Vladeta Jovovic, Sep 15 2002

			For n = 22 we have 4 such cycle types: [1, 1, 1, 3, 4, 5, 7], [1, 2, 3, 4, 5, 7], [3, 3, 4, 5, 7], [4, 5, 6, 7].

Alois P. Heinz, Table of n, a(n) for n = 0..180

Cf. A000793, A074859, A256067, A256554.

A000793 := proc(n) option remember; local l,p,i ; l := 1: p := combinat[partition](n): for i from 1 to combinat[numbpart](n) do if ilcm( p[i][j] $ j=1..nops(p[i])) > l then l := ilcm( p[i][j] $ j=1..nops(p[i])) ; fi: od: RETURN(l) ; end proc:
taylInv := proc(i,n) local resul,j,idiv,k ; resul := 1 ; idiv := numtheory[divisors](i) ; for k from 1 to nops(idiv) do j := op(k,idiv) ; resul := resul*taylor(1/(1-x^j),x=0,n+1) ; resul := convert(taylor(resul,x=0,n+1),polynom) ; od ; coeftayl(resul,x=0,n) ; end proc:
A074064 := proc(n) local resul,a793,dvs,i,k ; resul := 0: a793 := A000793(n) ; dvs := numtheory[divisors](a793) ; for k from 1 to nops(dvs) do i := op(k,dvs) ; resul := resul+numtheory[mobius](a793/i)*taylInv(i,n) ; od : RETURN(resul) ; end proc: # R. J. Mathar, Mar 30 2007

b[n_, i_] := b[n, i] = Module[{p}, p = If[i < 1, 1, Prime[i]]; If[n == 0 || i < 1, 1, Max[b[n, i-1], Table[p^j*b[n-p^j, i-1], {j, 1, Log[p, n] // Floor}]]]];
g[n_] := g[n] = b[n, If[n < 8, 3, PrimePi[Ceiling[1.328*Sqrt[n*Log[n] // Floor]]]]];
a[n_] := a[n] = SeriesCoefficient[Sum[MoebiusMu[g[n]/i]/Product[1-x^j, {j, Divisors[i]}], {i, Divisors[g[n]]}] + O[x]^(n+1), n];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 25 2017, after Alois P. Heinz *)

Coefficient of x^n in expansion of Sum_{i divides A000793(n)} mu(A000793(n)/i)*1/Product_{j divides i} (1-x^j).

More terms from R. J. Mathar, Mar 30 2007
More terms from Sean A. Irvine, Oct 04 2011
More terms from Alois P. Heinz, Mar 29 2015

A181844 Sum over all partitions of n of the LCM of the parts.

Original entry on oeis.org

1, 1, 3, 6, 12, 23, 38, 73, 118, 198, 318, 530, 819, 1298, 1974, 2975, 4516, 6698, 9980, 14550, 21186, 30304, 43503, 62030, 87908, 123292, 172543, 239720, 331688, 458198, 629376, 860332, 1168172, 1583176, 2138438, 2876283, 3859770, 5159886, 6863702, 9112356

Offset: 0

Peter Luschny, Dec 07 2010

nonn

Old name was: Row sums of A181842.

Alois P. Heinz, Table of n, a(n) for n = 0..188 (terms n=1..80 from Vincenzo Librandi)

Cf. A078392 (the same for GCD), A181843, A181842, A256067, A256553, A256554, A306956.

with(combstruct):
a181844 := proc(n) local k,L,l,R,part;
R := NULL; L := 0;
for k from 1 to n do
   part := iterstructs(Partition(n),size=k):
   while not finished(part) do
      l := nextstruct(part);
      L := L + ilcm(op(l));
   od;
od;
L end:
# second Maple program:
b:= proc(n, i, r) option remember; `if`(n=0, r, `if`(i<1, 0,
       b(n, i-1, r)+b(n-i, min(i, n-i), ilcm(i, r))))
    end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..42);  # Alois P. Heinz, Mar 18 2019

t[n_, k_] := LCM @@@ IntegerPartitions[n, {n - k + 1}] // Total; a[n_] := Sum[t[n, k], {k, 1, n}]; Table[a[n], {n, 1, 32}] (* Jean-François Alcover, Jul 26 2013 *)

a(n) = Sum_{k>=0} k * A256067(n,k) = Sum_{k>=0} A256553(n,k)*A256554(n,k). - Alois P. Heinz, Apr 02 2015

a(0)=1 prepended by Alois P. Heinz, Mar 29 2015

New name from Alois P. Heinz, Mar 18 2019

A074752 Number of combinatorially inequivalent cyclic subgroups of S_n of order 6. Number of partitions of n of order 6.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 12, 16, 19, 24, 29, 34, 40, 48, 54, 63, 72, 81, 91, 104, 114, 128, 142, 156, 171, 190, 205, 225, 245, 265, 286, 312, 333, 360, 387, 414, 442, 476, 504, 539, 574, 609, 645, 688, 724, 768, 812, 856, 901, 954, 999, 1053, 1107, 1161, 1216, 1280

Offset: 5

Vladeta Jovovic, Sep 28 2002

Two permutation groups are combinatorially equivalent iff they have the same cycle index. Order of partition is lcm of its parts.

Alois P. Heinz, Table of n, a(n) for n = 5..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,2,-1,-1,0,1,1,-1).

Column k=6 of A256067, A256554.

LinearRecurrence[{1,1,0,-1,-1,2,-1,-1,0,1,1,-1},{1,2,3,5,7,9,12,16,19,24,29,34},60] (* Harvey P. Dale, May 23 2020 *)

G.f.: x^5*(1+x-x^6)/((x-1)*(x^2-1)*(x^3-1)*(x^6-1)). More generally, g.f. for number of partitions of order d is Sum_{i divides d} mu(d/i)*1/Product_{j divides i} (1-x^j).

A256553 Triangle T(n,k) in which the n-th row contains the increasing list of distinct orders of degree-n permutations; n>=0, 1<=k<=A009490(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 10, 12, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 20, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 20, 21, 30

Offset: 0

Alois P. Heinz, Apr 01 2015

			Triangle T(n,k) begins:
  1;
  1;
  1, 2;
  1, 2, 3;
  1, 2, 3, 4;
  1, 2, 3, 4, 5, 6;
  1, 2, 3, 4, 5, 6;
  1, 2, 3, 4, 5, 6, 7, 10, 12;
  1, 2, 3, 4, 5, 6, 7,  8, 10, 12, 15;
  1, 2, 3, 4, 5, 6, 7,  8,  9, 10, 12, 14, 15, 20;
  1, 2, 3, 4, 5, 6, 7,  8,  9, 10, 12, 14, 15, 20, 21, 30;

Alois P. Heinz, Rows n = 0..60, flattened

Row sums give A060179.
Row lengths give A009490.
Last elements of rows give A000793.
Main diagonal gives A000027.
Cf. A181844, A256067, A256554.

b:= proc(n, i) option remember; `if`(n=0 or i=1, x,
      b(n, i-1)+(p-> add(coeff(p, x, t)*x^ilcm(t, i),
      t=1..degree(p)))(add(b(n-i*j, i-1), j=1..n/i)))
    end:
T:= n->(p->seq((h->`if`(h=0, [][], i))(coeff(p, x, i))
     , i=1..degree(p)))(b(n$2)):
seq(T(n), n=0..12);

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x,
     b[n, i - 1] + Function[p, Sum[Coefficient[p, x, t]*x^LCM[t, i],
     {t, 1, Exponent[p, x]}]][Sum[b[n - i*j, i - 1], {j, 1, n/i}]]];
T[n_] := Function[p, Table[Function[h, If[h == 0, Nothing, i]][
     Coefficient[p, x, i]], {i, 1, Exponent[p, x]}]][b[n, n]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jul 15 2021, after Alois P. Heinz *)

Sum_{k>=0} T(n,k)*A256554(n,k) = A181844(n).

T(n,k) = k for n>0 and 1<=k<=n.

cp's OEIS Frontend

A074761 Number of partitions of n of order n.

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A074064 Number of cycle types of degree-n permutations having the maximum possible order.

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A181844 Sum over all partitions of n of the LCM of the parts.

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A074752 Number of combinatorially inequivalent cyclic subgroups of S_n of order 6. Number of partitions of n of order 6.

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A256553 Triangle T(n,k) in which the n-th row contains the increasing list of distinct orders of degree-n permutations; n>=0, 1<=k<=A009490(n).

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